On the group of homeomorphisms on R: A revisit

Authors

  • Kamaludheen Ali Akbar Central University of Kerala
  • T. Mubeena University of Calicut

DOI:

https://doi.org/10.4995/agt.2022.16143

Keywords:

Group of homeomorphisms, Normal subgroups, Dynamical systems, Fixed points, Conjugacy, bounded functions

Abstract

 In this article, we prove that the group of all increasing homeomorphisms on R has exactly five normal subgroups, and the group of all homeomorphisms on R has exactly four normal subgroups. There are several results known about the group of homeomorphisms on R and about the group of increasing homeomorphisms on R ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. In this paper, we analyze this structure in some detail.

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Author Biographies

Kamaludheen Ali Akbar, Central University of Kerala

Department of Mathematics, School of Physical Sciences

T. Mubeena, University of Calicut

Department of Mathematics, School of Mathematics and Computational Science

References

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N. J. Fine and G. E. Schweigert, On the group of homeomorphisms of an arc, Annals of Mathematics 62 (1955), 237-253. https://doi.org/10.2307/1969678

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511755316

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, second edition, 1989.

I. N. Herstein, Topics in Algebra, John Wiley and Sons, 2nd Revised edition, 1975.

A. G. O'Farrell, Conjugacy, involutions, and reversibility for real homeomorphisms, Irish Math. Soc. Bulletin 54 (2004), 41-52. https://doi.org/10.33232/BIMS.0054.41.52

S. Ulam and J. von Neumann, On the group of homeomorphisms of the surface of the sphere, (abstract), Bull. Amer. Math. Soc. 53 (1947), 506.

J. V. Whittaker, Normal subgroups of some homeomorphism groups, Pacific J. Math. 10, no. 4 (1960), 1469-1478. https://doi.org/10.2140/pjm.1960.10.1469

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Published

2022-10-03

How to Cite

[1]
K. Ali Akbar and T. Mubeena, “On the group of homeomorphisms on R: A revisit”, Appl. Gen. Topol., vol. 23, no. 2, pp. 269–280, Oct. 2022.

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Articles

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